Analysis Football Gambling Study
Henrik
16 October, 2021
Preparation
Load & Prepare Data
We first load in the data from all participants and transform variables into correct type. We also perform some checks on the data to confirm that it matches our expectations (i.e., in size and structure).
Then we create the our DV, proportion bet, as follows (note that the initial endowment was £3): \[\texttt{prop_bet} = \frac{\texttt{amount bet}}{3}\]
The analysis is based on the following number of participants
## [1] 1003## # A tibble: 2 × 2
## condition n
## * <fct> <int>
## 1 None 501
## 2 Yellow 502Distribution of DV
Our DV clearly does not look normally distributed and shows some clear bump at certain prominent numbers.
Binomial confidence or credibility intervals for the probability to gamble at all:
## method x n mean lower upper
## 1 agresti-coull 608 1003 0.6061815 0.5755911 0.6359616
## 2 asymptotic 608 1003 0.6061815 0.5759439 0.6364190
## 3 bayes 608 1003 0.6060757 0.5758157 0.6362173
## 4 cloglog 608 1003 0.6061815 0.5752055 0.6356554
## 5 exact 608 1003 0.6061815 0.5751698 0.6365680
## 6 logit 608 1003 0.6061815 0.5755731 0.6359785
## 7 probit 608 1003 0.6061815 0.5756531 0.6360706
## 8 profile 608 1003 0.6061815 0.5757014 0.6361198
## 9 lrt 608 1003 0.6061815 0.5757042 0.6361101
## 10 prop.test 608 1003 0.6061815 0.5750905 0.6364488
## 11 wilson 608 1003 0.6061815 0.5755938 0.6359589Zero-One Inflated Beta Regression
We use a custom parameterization of a zero-one-inflated beta-regression model (see also here). The likelihood of the model is given by:
\[\begin{align} f(y) &= (1 - g) & & \text{if } y = 0 \\ f(y) &= g \times e & & \text{if } y = 1 \\ f(y) &= g \times (1 - e) \times \text{Beta}(a,b) & & \text{if } y \notin \{0, 1\} \\ a &= \mu \times \phi \\ b &= (1-\mu) \times \phi \end{align}\]
Where \(g\) is the zero inflation probability (zipp) and reflects the probability to gamble, \(e\) is the conditional one-inflation probability (coi) or conditional probability to gamble everything (i.e., conditional probability to have a value of one, if one gambles), \(\mu\) is the mean of the beta distribution (Intercept), and \(\phi\) is the precision of the beta distribution (phi). As we use Stan for modelling, we need to model on the real line and need appropriate link functions. For \phi the link is log (inverse is exp()), for all other parameters it is logit (inverse is plogis()).
We fit this model and add experimental condition as a factor to the three main model parameters (i.e., only the precision parameter is fixed across conditions). The following table provides the overview of the model and all model parameters and show good convergence.
## Family: zoib2
## Links: mu = logit; phi = log; zipp = logit; coi = logit
## Formula: new_prop ~ condition
## phi ~ 1
## zipp ~ condition
## coi ~ condition
## Data: part2 (Number of observations: 1003)
## Samples: 4 chains, each with iter = 26000; warmup = 1000; thin = 1;
## total post-warmup samples = 1e+05
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept -0.39 0.08 -0.53 -0.24 1.00 165326 79648
## phi_Intercept 1.80 0.10 1.60 1.98 1.00 156811 76279
## zipp_Intercept 0.41 0.09 0.23 0.59 1.00 162053 78870
## coi_Intercept 0.67 0.12 0.43 0.91 1.00 158057 76842
## conditionYellow -0.02 0.11 -0.24 0.20 1.00 153942 77003
## zipp_conditionYellow 0.05 0.13 -0.21 0.30 1.00 162122 76877
## coi_conditionYellow 0.28 0.18 -0.07 0.62 1.00 151240 75525
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).As a visual convergence check, we plot the density and trace plots for the four intercept parameters representing the no message condition or the overall mean (for phi).
We can also plot the three parameters showing the difference distribution of the message condition from the no message condition. These differences are given on the logit scale.
The model does not have any obvious problems, even without priors for the condition specific effects.
Posterior Predictive Checks
As expected the synthetic data generated from the model is able to adequately predict the two peaks at the boundary, but does not fully capture the peaks in between those. However, the mass of the distribution between the peaks seems to be at a similar location in both the observed and the synthetic data.
Model Estimates
We first give the table showing the posterior means and 95% CIs.
## # A tibble: 6 × 8
## # Groups: parameter [3]
## parameter condition estimate .lower .upper .width .point .interval
## <fct> <chr> <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 Gamble at all? None 0.600 0.557 0.643 0.95 mean qi
## 2 Gamble at all? Yellow 0.611 0.569 0.653 0.95 mean qi
## 3 Gamble everything? None 0.661 0.606 0.713 0.95 mean qi
## 4 Gamble everything? Yellow 0.719 0.668 0.768 0.95 mean qi
## 5 Proportion bet? None 0.405 0.370 0.441 0.95 mean qi
## 6 Proportion bet? Yellow 0.400 0.362 0.439 0.95 mean qiFor the zero-one inflated components, we can compare the model estimates with the data. Not unsurprisingly, they match quite well.
## # A tibble: 2 × 3
## condition gamble_at_all gamble_everything
## * <fct> <dbl> <dbl>
## 1 None 0.601 0.661
## 2 Yellow 0.612 0.720The following is the main results figure on the level of the message conditions.
Difference distribution versus no message condition
We can also focus and look at the difference distributions from the no message condition.
## # A tibble: 6 × 8
## # Groups: parameter [3]
## parameter condition estimate .lower .upper .width .point .interval
## <fct> <fct> <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 Gamble at all? " " 0.0108 -0.0285 0.0504 0.8 mean qi
## 2 Gamble everything? " " 0.0587 0.0112 0.106 0.8 mean qi
## 3 Proportion bet? " " -0.00442 -0.0386 0.0297 0.8 mean qi
## 4 Gamble at all? " " 0.0108 -0.0494 0.0712 0.95 mean qi
## 5 Gamble everything? " " 0.0587 -0.0144 0.132 0.95 mean qi
## 6 Proportion bet? " " -0.00442 -0.0565 0.0477 0.95 mean qiSame as a figure.
Difference Distributions with Possible Priors
The following plot shows all the difference distributions with overlayed density estimate (in black) and some possible prior distributions in colour (note again that the model did not actually include any priors). These priors are normal priors (who have a higher peak at 0 compared to Cauchy and t) with two different SDs.
We see that even with an extremely small prior with SD = 0.05 we still see evidence for no difference (i.e., black density estimate above the prior at 0) for all but “gamble evewrything” for which we see some evidence for a backfire effect. Note that a prior with SD of .05 means that we expect with 95% probability that the largest effect we observe is 2.50% on the response scale.
Model With Covariates
Plot of Relationships and BANOVA
The covariates do not differ between conditions (BF provide evidence for the null).
## Bayes factor analysis
## --------------
## [1] Intercept only : 5.980854 ±0.01%
##
## Against denominator:
## pgsi ~ condition
## ---
## Bayes factor type: BFlinearModel, JZS## Bayes factor analysis
## --------------
## [1] Intercept only : 2.146103 ±0%
##
## Against denominator:
## motives ~ condition
## ---
## Bayes factor type: BFlinearModel, JZSThe following table shows descriptives for the covariates:
## # A tibble: 2 × 5
## condition pgsi_mean pgsi_sd motives_mean motives_sd
## * <fct> <dbl> <dbl> <dbl> <dbl>
## 1 None 2.12 3.84 5.23 3.66
## 2 Yellow 1.82 3.45 4.77 3.73The following plot shows the relationships of the covariates among each other and with proportion bet.
The figures show some relationships which can also be seen when just looking at the correlation.
##
## Pearson's product-moment correlation
##
## data: part2$pgsi and part2$motives
## t = 12.228, df = 1001, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## 0.3054271 0.4131911
## sample estimates:
## cor
## 0.3605115##
## Pearson's product-moment correlation
##
## data: part2$pgsi and part2$new_prop
## t = 0.70989, df = 1001, p-value = 0.4779
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## -0.03952346 0.08421501
## sample estimates:
## cor
## 0.02243168##
## Pearson's product-moment correlation
##
## data: part2$motives and part2$new_prop
## t = 6.4399, df = 1001, p-value = 1.853e-10
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## 0.1392754 0.2581689
## sample estimates:
## cor
## 0.1994561The figure below provides an alternative visualisation of the relationships between covariates and betting behaviour. In particular, participants were categorized into one of three experimental betting behavior groups: participants who did not bet at all (“none”, 39% of participants), participants who bet some of their money (19% of participants), and participants who bet “all” of their money (42%). For both gambling scales we see a positive relationship between the betting behavior group and the gambling score. Participants who bet more have on average higher scores on the two gambling scales.
##
## none some all
## 0.3938185 0.1874377 0.4187438
This is supported by Bayesian ANOVAs for the enhancement motives scale, with a Bayes factor of over 700,000 for the effect of betting behavior group on motives score. However, there was evidence for a null effect (BF around 50 for the null) for the PGSI effect even though it showed descriptively the same pattern.
Furthermore, the effect on the motive score was not moderated by gambling message condition. The evidence for the absence of a main effect of condition was ambigous (around 1), but there was substantial evidence for the absence of the interaction (BF > 30).
## Bayes factor analysis
## --------------
## [1] condition : 0.1672002 ±0.01%
## [2] gamble_cat : 0.01975068 ±0.03%
## [3] condition + gamble_cat : 0.003681872 ±5.3%
## [4] condition + gamble_cat + condition:gamble_cat : 0.0001180591 ±3.27%
##
## Against denominator:
## Intercept only
## ---
## Bayes factor type: BFlinearModel, JZS## denominator
## numerator condition gamble_cat condition + gamble_cat condition + gamble_cat + condition:gamble_cat
## gamble_cat 0.1181259 1 5.364305 167.2949## Bayes factor analysis
## --------------
## [1] condition : 0.465961 ±0%
## [2] gamble_cat : 732922.3 ±0.03%
## [3] condition + gamble_cat : 510898 ±1.35%
## [4] condition + gamble_cat + condition:gamble_cat : 21425.99 ±2.68%
##
## Against denominator:
## Intercept only
## ---
## Bayes factor type: BFlinearModel, JZS## denominator
## numerator condition gamble_cat condition + gamble_cat condition + gamble_cat + condition:gamble_cat
## gamble_cat 1572927 1 1.434577 34.20717Model with Main Effects
## Family: zoib2
## Links: mu = logit; phi = log; zipp = logit; coi = logit
## Formula: new_prop ~ condition + pgsi_c + motives_c
## phi ~ 1
## zipp ~ condition + pgsi_c + motives_c
## coi ~ condition + pgsi_c + motives_c
## Data: part2 (Number of observations: 1003)
## Samples: 4 chains, each with iter = 26000; warmup = 1000; thin = 1;
## total post-warmup samples = 1e+05
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept -0.41 0.08 -0.56 -0.27 1.00 204638 78875
## phi_Intercept 1.83 0.10 1.63 2.02 1.00 199577 74920
## zipp_Intercept 0.41 0.09 0.22 0.59 1.00 223505 73569
## coi_Intercept 0.64 0.12 0.40 0.88 1.00 220411 77103
## conditionYellow 0.02 0.11 -0.19 0.24 1.00 199808 77825
## pgsi_c -0.02 0.02 -0.05 0.01 1.00 195904 82129
## motives_c 0.04 0.02 0.01 0.08 1.00 187119 82114
## zipp_conditionYellow 0.09 0.13 -0.17 0.35 1.00 214675 71246
## zipp_pgsi_c -0.04 0.02 -0.08 -0.00 1.00 187697 84188
## zipp_motives_c 0.12 0.02 0.08 0.16 1.00 181763 86458
## coi_conditionYellow 0.30 0.18 -0.05 0.65 1.00 222578 75461
## coi_pgsi_c 0.01 0.03 -0.04 0.06 1.00 184080 79982
## coi_motives_c 0.05 0.03 0.00 0.11 1.00 186304 81217
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).As a visual convergence check, we plot the density and trace plots for the four intercept parameters representing the no message condition or the overall mean (for phi).
We can also plot the three parameters showing the difference distribution of the message condition from the no message condition. These differences are given on the logit scale.
We also check the chains for the PGSI and motives scales. 
Model with Main Effects (only PGSI)
## Family: zoib2
## Links: mu = logit; phi = log; zipp = logit; coi = logit
## Formula: new_prop ~ condition + pgsi_c
## phi ~ 1
## zipp ~ condition + pgsi_c
## coi ~ condition + pgsi_c
## Data: part2 (Number of observations: 1003)
## Samples: 4 chains, each with iter = 26000; warmup = 1000; thin = 1;
## total post-warmup samples = 1e+05
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept -0.39 0.08 -0.54 -0.24 1.00 188685 77974
## phi_Intercept 1.79 0.10 1.60 1.98 1.00 184659 77602
## zipp_Intercept 0.41 0.09 0.23 0.59 1.00 190736 77024
## coi_Intercept 0.67 0.12 0.43 0.91 1.00 196512 76475
## conditionYellow -0.02 0.11 -0.24 0.20 1.00 192125 76321
## pgsi_c -0.01 0.02 -0.04 0.02 1.00 194361 75283
## zipp_conditionYellow 0.05 0.13 -0.21 0.30 1.00 194240 76896
## zipp_pgsi_c 0.00 0.02 -0.03 0.04 1.00 194850 79082
## coi_conditionYellow 0.28 0.18 -0.06 0.63 1.00 194609 77274
## coi_pgsi_c 0.03 0.03 -0.02 0.08 1.00 190892 75247
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).As a visual convergence check, we plot the density and trace plots for the four intercept parameters representing the no message condition or the overall mean (for phi).
We can also plot the three parameters showing the difference distribution of the message condition from the no message condition. These differences are given on the logit scale.
We also check the chains for the PGSI and motives scales. 
Model with Main Effects (only Motives)
## Family: zoib2
## Links: mu = logit; phi = log; zipp = logit; coi = logit
## Formula: new_prop ~ condition + motives_c
## phi ~ 1
## zipp ~ condition + motives_c
## coi ~ condition + motives_c
## Data: part2 (Number of observations: 1003)
## Samples: 4 chains, each with iter = 26000; warmup = 1000; thin = 1;
## total post-warmup samples = 1e+05
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept -0.40 0.08 -0.55 -0.26 1.00 181035 77448
## phi_Intercept 1.82 0.10 1.63 2.01 1.00 176269 75403
## zipp_Intercept 0.40 0.09 0.22 0.59 1.00 188942 77503
## coi_Intercept 0.64 0.12 0.40 0.88 1.00 188149 76095
## conditionYellow 0.02 0.11 -0.20 0.24 1.00 185964 78765
## motives_c 0.04 0.02 0.01 0.07 1.00 184365 76536
## zipp_conditionYellow 0.10 0.13 -0.16 0.35 1.00 190303 77932
## zipp_motives_c 0.10 0.02 0.07 0.14 1.00 187308 79873
## coi_conditionYellow 0.30 0.18 -0.04 0.65 1.00 186241 76110
## coi_motives_c 0.06 0.02 0.01 0.11 1.00 183293 78027
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).As a visual convergence check, we plot the density and trace plots for the four intercept parameters representing the no message condition or the overall mean (for phi).
We can also plot the three parameters showing the difference distribution of the message condition from the no message condition. These differences are given on the logit scale.
We also check the chains for the PGSI and motives scales. 
Model With Interactions
## Family: zoib2
## Links: mu = logit; phi = log; zipp = logit; coi = logit
## Formula: new_prop ~ condition * (pgsi_c + motives_c)
## phi ~ 1
## zipp ~ condition * (pgsi_c + motives_c)
## coi ~ condition * (pgsi_c + motives_c)
## Data: part2 (Number of observations: 1003)
## Samples: 4 chains, each with iter = 26000; warmup = 1000; thin = 1;
## total post-warmup samples = 1e+05
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept -0.40 0.07 -0.55 -0.26 1.00 158995 75025
## phi_Intercept 1.88 0.10 1.68 2.07 1.00 152151 75986
## zipp_Intercept 0.41 0.09 0.23 0.59 1.00 156883 74314
## coi_Intercept 0.66 0.12 0.42 0.90 1.00 161327 75990
## conditionYellow 0.04 0.11 -0.18 0.25 1.00 157775 75353
## pgsi_c -0.05 0.02 -0.09 -0.00 1.00 122834 79156
## motives_c 0.01 0.02 -0.03 0.05 1.00 125734 81613
## conditionYellow:pgsi_c 0.03 0.03 -0.03 0.09 1.00 117220 80986
## conditionYellow:motives_c 0.09 0.03 0.02 0.15 1.00 123975 84181
## zipp_conditionYellow 0.10 0.13 -0.16 0.36 1.00 162568 72966
## zipp_pgsi_c -0.05 0.03 -0.10 -0.00 1.00 110491 80733
## zipp_motives_c 0.12 0.03 0.06 0.17 1.00 108240 79092
## zipp_conditionYellow:pgsi_c 0.03 0.04 -0.05 0.11 1.00 111500 83582
## zipp_conditionYellow:motives_c 0.01 0.04 -0.07 0.09 1.00 106528 80811
## coi_conditionYellow 0.29 0.18 -0.06 0.64 1.00 160553 74693
## coi_pgsi_c 0.02 0.04 -0.05 0.10 1.00 111410 74118
## coi_motives_c 0.03 0.04 -0.04 0.10 1.00 110727 82005
## coi_conditionYellow:pgsi_c -0.02 0.06 -0.13 0.09 1.00 111916 80660
## coi_conditionYellow:motives_c 0.05 0.05 -0.05 0.16 1.00 110909 81778
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).As a visual convergence check, we plot the density and trace plots for the four intercept parameters representing the no message condition or the overall mean (for phi).
We can also plot the three parameters showing the difference distribution of the yellow message condition from the no messafe condition. These differences are given on the logit scale.
Model With Interactions (only PGSI)
## Family: zoib2
## Links: mu = logit; phi = log; zipp = logit; coi = logit
## Formula: new_prop ~ condition * (pgsi_c)
## phi ~ 1
## zipp ~ condition * (pgsi_c)
## coi ~ condition * (pgsi_c)
## Data: part2 (Number of observations: 1003)
## Samples: 4 chains, each with iter = 26000; warmup = 1000; thin = 1;
## total post-warmup samples = 1e+05
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept -0.39 0.08 -0.54 -0.24 1.00 142224 78453
## phi_Intercept 1.82 0.10 1.62 2.00 1.00 128555 79050
## zipp_Intercept 0.41 0.09 0.23 0.59 1.00 131718 75828
## coi_Intercept 0.67 0.12 0.43 0.91 1.00 131706 80557
## conditionYellow -0.01 0.11 -0.23 0.21 1.00 132020 77502
## pgsi_c -0.04 0.02 -0.09 -0.00 1.00 88573 71395
## conditionYellow:pgsi_c 0.06 0.03 0.01 0.12 1.00 88715 74078
## zipp_conditionYellow 0.05 0.13 -0.20 0.31 1.00 129867 77496
## zipp_pgsi_c -0.01 0.02 -0.06 0.04 1.00 93322 76809
## zipp_conditionYellow:pgsi_c 0.04 0.04 -0.04 0.11 1.00 92163 76720
## coi_conditionYellow 0.28 0.18 -0.07 0.63 1.00 117684 74900
## coi_pgsi_c 0.03 0.04 -0.04 0.11 1.00 86101 71033
## coi_conditionYellow:pgsi_c -0.01 0.05 -0.11 0.10 1.00 85775 75271
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).As a visual convergence check, we plot the density and trace plots for the four intercept parameters representing the no message condition or the overall mean (for phi).
We can also plot the three parameters showing the difference distribution of the yellow message condition from the no message condition. These differences are given on the logit scale.
Is the overall effect of PGSI credibly negative?
## # A tibble: 2 × 7
## condition .value .lower .upper .width .point .interval
## <fct> <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 None -0.0418 -0.0870 -0.00108 0.95 median qi
## 2 Yellow 0.0221 -0.0187 0.0622 0.95 median qi## # A tibble: 1 × 6
## Overall .lower .upper .width .point .interval
## <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 -0.00999 -0.0403 0.0189 0.95 median qiModel With Interactions (only Motives)
## Family: zoib2
## Links: mu = logit; phi = log; zipp = logit; coi = logit
## Formula: new_prop ~ condition * (motives_c)
## phi ~ 1
## zipp ~ condition * (motives_c)
## coi ~ condition * (motives_c)
## Data: part2 (Number of observations: 1003)
## Samples: 4 chains, each with iter = 26000; warmup = 1000; thin = 1;
## total post-warmup samples = 1e+05
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept -0.39 0.07 -0.54 -0.25 1.00 171436 77328
## phi_Intercept 1.86 0.10 1.66 2.05 1.00 142562 78531
## zipp_Intercept 0.40 0.09 0.22 0.58 1.00 152884 78418
## coi_Intercept 0.65 0.12 0.41 0.89 1.00 144527 76552
## conditionYellow 0.03 0.11 -0.19 0.24 1.00 146423 79322
## motives_c 0.00 0.02 -0.04 0.04 1.00 93427 83551
## conditionYellow:motives_c 0.09 0.03 0.03 0.15 1.00 92740 81180
## zipp_conditionYellow 0.10 0.13 -0.16 0.37 1.00 147099 77946
## zipp_motives_c 0.09 0.03 0.04 0.15 1.00 82058 80255
## zipp_conditionYellow:motives_c 0.02 0.04 -0.05 0.09 1.00 83602 79198
## coi_conditionYellow 0.29 0.18 -0.06 0.65 1.00 137634 78052
## coi_motives_c 0.04 0.03 -0.03 0.10 1.00 88051 81948
## coi_conditionYellow:motives_c 0.04 0.05 -0.05 0.14 1.00 89138 81054
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).As a visual convergence check, we plot the density and trace plots for the four intercept parameters representing the no message condition or the overall mean (for phi).
We can also plot the three parameters showing the difference distribution of the yellow message condition from the no message condition. These differences are given on the logit scale.
Is the overall effect of Motives credibly negative?
## # A tibble: 2 × 7
## condition .value .lower .upper .width .point .interval
## <fct> <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 None 0.000446 -0.0382 0.0386 0.95 median qi
## 2 Yellow 0.0878 0.0402 0.137 0.95 median qi## # A tibble: 1 × 6
## Overall .lower .upper .width .point .interval
## <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 0.0441 0.0134 0.0751 0.95 median qiRiskiness
For the analysis of riskiness, we had to exclude all participants that did not place a single bet. This led to the following N:
## # A tibble: 2 × 2
## condition n
## * <fct> <int>
## 1 None 301
## 2 Yellow 307The distribution of riskiness is:
It turns out that there are two participants with a scaled riskiness of zero, both in the treatment condition. And no participants with a riskiness of 1.
## # A tibble: 2 × 3
## condition n_zero n_one
## * <fct> <int> <int>
## 1 None 0 0
## 2 Yellow 2 0Beta-Regression
We run the first analysis, using a beta-regression model, after excluding the two zeros.
The model does not show any obvious problems. In addition, we can see that the 95%-CI for the gambling message specific effect includes 0.
## Family: beta
## Links: mu = logit; phi = log
## Formula: scaled_risk ~ condition
## phi ~ 1
## Data: risk3 (Number of observations: 606)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept -3.07 0.07 -3.20 -2.93 1.00 2148 2383
## phi_Intercept 2.30 0.07 2.15 2.45 1.00 1880 2484
## conditionYellow -0.10 0.08 -0.25 0.05 1.00 2676 2545
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
The data and the posterior predictive distribution look good.
Zero-Inflated Beta-Regression
We re-run the analysis, using a zero-inflated beta-regression model.
The model also does not show any obvious problems. As above, the 95% CIs of the condition effects contain 0 indicated no evidence that the warning label affects the riskiness of gambles.
## Family: zero_inflated_beta
## Links: mu = logit; phi = log; zi = logit
## Formula: scaled_risk ~ condition
## zi ~ condition
## phi ~ 1
## Data: risk2 (Number of observations: 608)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept -3.06 0.07 -3.20 -2.93 1.00 2189 2274
## phi_Intercept 2.30 0.07 2.16 2.44 1.00 1954 1771
## zi_Intercept -7.67 2.17 -13.16 -4.82 1.00 1510 1087
## conditionYellow -0.10 0.08 -0.26 0.06 1.00 2610 2452
## zi_conditionYellow 2.67 2.27 -0.71 8.28 1.00 1466 1165
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
As expected, the data and the posterior predictive distribution also look good.
